{"title":"Spectrum of Lévy–Khintchine random laplacian matrices","doi":"10.1007/s10959-023-01275-4","article_type":"original","external_id":{"isi":["001038341000001"],"arxiv":["2210.07927"]},"article_processing_charge":"Yes (via OA deal)","department":[{"_id":"LaEr"}],"publisher":"Springer Nature","author":[{"last_name":"Campbell","full_name":"Campbell, Andrew J","id":"582b06a9-1f1c-11ee-b076-82ffce00dde4","first_name":"Andrew J"},{"first_name":"Sean","full_name":"O’Rourke, Sean","last_name":"O’Rourke"}],"date_created":"2023-08-06T22:01:13Z","oa":1,"abstract":[{"lang":"eng","text":"We consider the spectrum of random Laplacian matrices of the form Ln=An−Dn where An\r\n is a real symmetric random matrix and Dn is a diagonal matrix whose entries are equal to the corresponding row sums of An. If An is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution, the empirical spectral measure of Ln is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices An with independent entries (up to symmetry) whose row sums converge to a purely non-Gaussian infinitely divisible distribution, which fall into the class of Lévy–Khintchine random matrices first introduced by Jung [Trans Am Math Soc, 370, (2018)]. Our main result shows that the empirical spectral measure of Ln  converges almost surely to a deterministic limit. A key step in the proof is to use the purely non-Gaussian nature of the row sums to build a random operator to which Ln converges in an appropriate sense. This operator leads to a recursive distributional equation uniquely describing the Stieltjes transform of the limiting empirical spectral measure."}],"publication_identifier":{"eissn":["1572-9230"],"issn":["0894-9840"]},"publication":"Journal of Theoretical Probability","type":"journal_article","date_published":"2023-07-26T00:00:00Z","citation":{"ista":"Campbell AJ, O’Rourke S. 2023. Spectrum of Lévy–Khintchine random laplacian matrices. Journal of Theoretical Probability.","ama":"Campbell AJ, O’Rourke S. Spectrum of Lévy–Khintchine random laplacian matrices. <i>Journal of Theoretical Probability</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s10959-023-01275-4\">10.1007/s10959-023-01275-4</a>","mla":"Campbell, Andrew J., and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random Laplacian Matrices.” <i>Journal of Theoretical Probability</i>, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s10959-023-01275-4\">10.1007/s10959-023-01275-4</a>.","short":"A.J. Campbell, S. O’Rourke, Journal of Theoretical Probability (2023).","ieee":"A. J. Campbell and S. O’Rourke, “Spectrum of Lévy–Khintchine random laplacian matrices,” <i>Journal of Theoretical Probability</i>. Springer Nature, 2023.","chicago":"Campbell, Andrew J, and Sean O’Rourke. “Spectrum of Lévy–Khintchine Random Laplacian Matrices.” <i>Journal of Theoretical Probability</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s10959-023-01275-4\">https://doi.org/10.1007/s10959-023-01275-4</a>.","apa":"Campbell, A. J., &#38; O’Rourke, S. (2023). Spectrum of Lévy–Khintchine random laplacian matrices. <i>Journal of Theoretical Probability</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10959-023-01275-4\">https://doi.org/10.1007/s10959-023-01275-4</a>"},"acknowledgement":"The first author thanks Yizhe Zhu for pointing out reference [30]. We thank David Renfrew for comments on an earlier draft. We thank the anonymous referee for a careful reading and helpful comments.\r\nOpen access funding provided by Institute of Science and Technology (IST Austria).","ddc":["510"],"year":"2023","scopus_import":"1","status":"public","oa_version":"Published Version","language":[{"iso":"eng"}],"isi":1,"tmp":{"short":"CC BY (4.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"month":"07","day":"26","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"13975","main_file_link":[{"url":"https://doi.org/10.1007/s10959-023-01275-4","open_access":"1"}],"has_accepted_license":"1","date_updated":"2023-12-13T12:00:50Z","publication_status":"epub_ahead"}